Fact:Start with $V$ a subspace of $\mathbb R^n$. Take the set of all supports of vectors in $V$. Throw out $\emptyset$. You now have the dependent sets of some matroid.

Not sure you believe me? Or just want to get your hands on the *independent* sets? (If you tell people that you have a matroid, they always ask you for the independent sets. How can you blame them?) Okay, no problem.

Choose a basis for $V^\perp$ and take those vectors as the rows of a matrix $A$. The sets of indices corresponding to linearly independent columns give the independent sets of the matroid. (A set of indices corresponds to linearly *dependent* columns, of course, exactly when it can be matched up with nonzero scalars so as to give a vector orthogonal to every row, i.e., a vector in $V$. This also shows that it doesn't matter which basis you choose.)

It follows from the above that *the matroids that arise in this fashion are precisely the ones that are representable* over $\mathbb R$.

I like this way of thinking about representability. But I've never seen it presented this way in the literature. Why not? (Maybe it has to do with how I seem to be relying on a nice Euclidean inner product?) This leads me to a couple of questions.

- Is the fact I stated above presented in any reference anywhere? I would greatly appreciate it if anyone could point me in the direction of one.
- Is it the case that the fact I stated above remains true when $\mathbb R$ is replaced by any field? If so, does it characterize representability over any field?

**EDIT:** I just realized that it's not difficult to prove that the above fact does hold over any field; I believe one can simply show that those support sets which are minimal with respect to $\subseteq$ satisfy the circuit axioms, and that's enough. So my main question here is whether or not this stills serves to characterize representability over other fields.